Neil led Norwich City to Premier League promotion in 2015 and will be tasked with achieving the same feat at The Hawthorns.
Telegraph Sport reports that West Brom are preparing an official approach to be made after their Championship clash this weekend.
Albion are said to be willing to pay £650,000 in compensation to bring in the Scot, who had led Preston to play-off contention this season and last.
Darren Moore was sacked by the Baggies in March and Neil heads their list of candidates for a permanent replacement ahead of Slavisa Jokanovic.
However, Preston are unlikely to let the 37-year-old go easily.
West Brom are currently fourth in the Championship standings, nine points behind Leeds in the second automatic promotion spot.
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